# Full text of "Treatise On Analysis Vol-Ii"

## See other formats

```84        XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

of a single point is nowhere dense unless it is open, i.e., unless a is isolated
in E (3.10.10).

In a topological group G, every closed subgroup H which is not open is
nowhere dense (12.8.7).

In a topological space E, a subset B is said to be meager if it is the union of
at most denumerably many nowhere-dense sets. Since the closure of a nowhere-
dense set is nowhere dense by definition, Baire's theorem can be restated in the
form that, in a space E satisfying the conditions of (12.16.1), the complement
of a meager set is dense in E.

For example, in a Frechet space E, a denumerable union of closed linear
varieties, each distinct from E, i.e., sets of the form an + Vrt, where Vn ^ E is a
closed vector subspace of E, has a dense complement in E. For it follows from

(12.13.1) that a vector subspace V of a topological vector space E cannot be
open in E unless V = E. In particular, a denumerable set in a Frechet space is
meager, but it should be noted that such a set may also be dense (if the space is
separable).

(12.16.2)    Let Ebea topological space in which every point has a neighborhood
homeomorphic to a complete metric space, and let u be a lower semicontinuous
function on E. Ifu(x) < -|- oo for each x e E, then given any nonempty open set
U ITJ E there exists a nonempty open set V c U such that sup u(x) < + oo.

xeV

It is enough to prove the result when U = E. For each integer n > 0, let Fn
be the set of all x e E such that u(x) £ n. By hypothesis, FM is a closed set
(12.7.2), and E is the union of the Fn; hence at least one of the Fw is not no-
where dense (12.16.1), and therefore contains a nonempty open set. Q.E.D.

Notice that under the hypotheses of (12.16.2) it can happen that

sup u(x) =4-00.

xeE

Consider for example the real-valued function on R which is equal to 0
at x - 0, and equal to l/x2 when x ^ 0.

In particular:

(12.16.3)    In a Frechet space E, every lower semicontinuous seminorm is con-
tinuous.

By definition, a seminorm/? on E is finite at all points of E. Ifp is lower
semicontinuous, it follows from (12.16.2) that there exists a point x0 e E, ace or a locally compact metrizable space (by virtue
```